A note on degenerate multi-poly-Genocchi polynomials
aa r X i v : . [ m a t h . N T ] M a y A NOTE ON DEGENERATE MULTI-POLY-GENOCCHIPOLYNOMIALS
TAEKYUN KIM , DAE SAN KIM , HAN YOUNG KIM , AND JONGKYUM KWON ∗ Abstract.
In this paper, we introduce the degenerate multiple polyexpo-nential functions which are multiple versions of the degenerate modifiedpolyexponential functions. Then we consider the degenerate multi-poly-Genocchi polynomials which are defined by using those functions and in-vestigate explicit expressions and some properties for those polynomials.
1. Introduction
It was Carlitz who initiated the study of degenerate versions of some specialnumbers and polynomials, namely the degenerate Bernoulli and Euler polyno-mials and numbers [1]. In recent years, studying degenerate versions of somespecial polynomials and numbers regained interests of many mathematicians,and quite a few interesting results were discovered [6,7,9,11-15,17]. The polyex-ponential functions were first introduced by Hardy in [3,4] and rediscovered byKim [11,14], as inverses to the polylogarithm functions. Recently, the degener-ate polyexponential functions, which are degenerate versions of polyexponentialfunctions, were introduced in [11], and some of their properties were investi-gated. Furthermore, the so-called new type degenerate Bell polynomials wereintroduced, and some identities connecting these polynomials to the degeneratepolyexponential functions were found in [11]. In [12], the (modified) polyex-ponential functions were used in order to define the degenerate poly-Bernoullipolynomials, and several explicit expressions about those polynomials and someidentities involving them were derived. In [14], the degenerate (modified) poly-exponential functions were introduced, and the degenerate type 2 poly-Bernoullinumbers and polynomials were defined by means of those functions. In addition,several explicit expressions and some identities for those numbers and polyno-mials were deduced.
Mathematics Subject Classification.
Key words and phrases. degenerate multiple polyexponential function; degenerate multi-poly-Genocchi polynomials.* is corresponding author. , DAE SAN KIM , HAN YOUNG KIM , AND JONGKYUM KWON ∗ In this paper, we intorduce the degenerate multiple polyexponential func-tions. These are multiple versions of the degenerate modified polyexponentialfunctions. Then we define the degenerate multi-poly-Gennocchi polynomials bymeans of those functions. We derive some explicit expressions for the degen-erate multi-poly-Gennocchi polynomials and certain properties related to thosepolynomials.We recall that, for all k ∈ Z , the polylogarithm functions are defined by Li k ( x ) = ∞ X k =1 x k n k , ( | x | < , (see [16 , . (1.1)The polyexponential functions were studied by Hardy in [3,4].Recently, a slightly different version of those functions, which are called themodified polyexponential functions, are defined as an inverse to polylogarithmfunctions by Ei k ( x ) = ∞ X n =1 x n ( n − n k , ( k ∈ Z ) , (see [5 , − , , , . (1.2)When k = 1, by (1.2), we get Ei ( x ) = ∞ X n =1 x n n ! = e x − . (1.3)In [14] (see also [11]), the degenerate modified polyexponential functions aredefined by Ei k,λ ( x ) = ∞ X n =1 (1) n,λ ( n − n k x n , ( λ ∈ R ) , (1.4)where ( x ) ,λ = 1, ( x ) n,λ = x ( x − λ ) · · · ( x − ( n − λ ), ( n ≥
1) .From (1.4), we note that Ei ,λ ( x ) = ∞ X n =1 (1) n,λ n ! x n = e λ ( x ) − . (1.5)The degenerate exponential functions are given by e xλ ( t ) = (1 + λt ) xλ , e λ ( t ) = e λ ( t ), and the degenerate logarithm functions are defined by log λ ( t ) = λ ( t λ − e λ ( t ).The degenerate poly–Genocchi polynomials are defined by2 Ei k,λ (log λ (1 + t )) e λ ( t ) + 1 e xλ ( t ) = ∞ X n =0 g ( k ) n,λ ( x ) t n n ! , ( k ∈ Z ) . (1.6) NOTE ON DEGENERATE MULTI-POLY-GENOCCHI POLYNOMIALS 3
For x = 0, g ( k ) n,λ = g ( k ) n,λ (0) are called the degenerate poly–Genochhi numbers.Here g (1) n,λ ( x ) = G n,λ ( x ) are the degenerate Genocchi polynomials given by2 te λ ( t ) + 1 e xλ ( t ) = ∞ X n =0 G n,λ ( x ) t n n ! , (see [12 , . (1.7)More generally, for r ∈ N , the Genocchi polynomials of order r are defined by (cid:18) te λ ( t ) + 1 (cid:19) r e xλ ( t ) = ∞ X n =0 G ( r ) n,λ ( x ) t n n ! , (see [13]) . (1.8)From (1.6), we note that g ( k ) n,λ ( x ) = n X l =0 (cid:18) nl (cid:19) g ( k ) l,λ ( x ) n − l,λ , ( n ≥ . (1.9)We will need the Carlitz’s degenerate Euler polynomials E ( r ) l,λ ( x ) of order r given by (cid:18) e λ ( t ) + 1 (cid:19) r e xλ ( t ) = ∞ X n =0 E ( r ) n,λ ( x ) t n n ! . (1.10)For k , k , · · · , k r ∈ Z , we define the degenerate multiple polyexponentialfunction as Ei k ,k , ··· ,k r ,λ ( x ) = X 2. Degenerate multi-poly-Genocchi polynomials For λ ∈ R , the degenerate Stirling numbers of the first kind are defined by( x ) n = n X l =0 S ,λ ( n, l )( x ) l,λ , ( n ≥ , (see [1 , , − , (2.1)where ( x ) = 1, ( x ) n = x ( x − · · · ( x − n + 1) , ( n ≥ k ! (log λ (1 + t )) k = ∞ X n = k S ,λ ( n, k ) x n n ! , ( k ≥ , (2.2)and that lim λ → S ,λ = S ( n, l ), where S ( n, l ) is the Stirling number of the firstkind. TAEKYUN KIM , DAE SAN KIM , HAN YOUNG KIM , AND JONGKYUM KWON ∗ For k , k , · · · , k r ∈ Z , we consider the degenerate mulit–poly–Genocchi poly-nomials which are given by2 r Ei k ,k , ··· ,k r,λ (log λ (1+ t )) ( e λ ( t ) + 1) r e xλ ( t ) = ∞ X n =0 g ( k ,k , ··· ,k r ) n,λ ( x ) t n n ! . (2.3)For x = 0, g ( k ,k , ··· ,k r ) n,λ = g ( k ,k , ··· ,k r ) n,λ (0) are called the degenerate multi–poly–Genocchi numbers.From (2.3), we note that g ( k , ··· ,k r ) n,λ ( x ) = n X l =0 (cid:18) nl (cid:19) g ( k ,k , ··· ,k r ) l,λ ( x ) n − l,λ , ( n ≥ . (2.4)From (2.3), we have ∞ X n =0 g ( k ,k , ··· ,k r ) n,λ ( x ) t n n != (cid:18) e λ ( t ) + 1 (cid:19) r e xλ ( t ) X Theorem 2.1. For k , k , . . . , k r ∈ Z , and n, r ∈ N with n ≥ r , we have g ( k ,k , ··· ,k r ) n,λ ( x )= n − r X l =0 (cid:18) nl (cid:19) E ( r ) l,λ ( x ) X For k , k , . . . , k r ∈ Z , and n, r ∈ N with n ≥ r , we have g ( k , ··· ,k r ) n,λ ( x ) = n − r X l =0 (cid:0) nl (cid:1) r ! (cid:0) l + rl (cid:1) G ( r ) l + r,λ ( x ) × X 12 2 e λ ( t ) + 1 (cid:19) r Ei k , ··· ,k r ,λ (log λ (1 + t ))= ∞ X m =0 (cid:18) r X l =0 (cid:18) rl (cid:19) ( − l r − l E ( l ) m,λ (cid:19) t m m ! × X Theorem 2.3. For k , k , . . . , k r ∈ Z , and n, r ∈ N with n ≥ r , we have g ( k , ··· ,k r ) n,λ ( r )= n − r X m =0 r X l =0 X For k , k , . . . , k r ∈ Z , and any nonnegative integer n , wehave g ( k , ··· ,k r ) n,λ ( x + y ) = n X l =0 (cid:18) nl (cid:19) g ( k , ··· ,k r ) l,λ ( x )( y ) n − l,λ . 3. Conclusion As we mentioned in the Introduction, studying various versions of some spe-cial polynomials and numbers has drawn attention of some mathematicians, andmany interesting results about those polynomials and numbers have been ob-tained. To state a few, these include the degenerate Stirling numbers of the first NOTE ON DEGENERATE MULTI-POLY-GENOCCHI POLYNOMIALS 7 and second kinds, degenerate central factorial numbers of the second kind, degen-erate Bernoulli numbers of the second kind, degenerate Bernstein polynomials,degenerate Bell numbers and polynomials, degenerate central Bell numbers andpolynomials, degenerate complete Bell polynomials and numbers, degenerateCauchy numbers, and so on (see [11,12,14] and the references therein). We notehere that the study has been carried out by using several different tools, such asgenerating functions, combinatorial methods, p -adic analysis, umbral calculus,differential equations, probability theory and so on.In this paper, we intorduced the degenerate multiple polyexponential func-tions which are multiple versions of the degenerate modified polyexponentialfunctions. Then we defined the degenerate multi-poly-Gennocchi polynomialsby means of those functions. In addition, we derived some explicit expressionsfor the degenerate multi-poly-Gennocchi polynomials and certain properties re-lated to those polynomials.It is one of our future projects to continue this line of research, namely tostudy degenerate versions of certain special polynomials and numbers and to findtheir applications in physics, science and engineering as well as in mathematics. Availability of data and material: Not applicable. Competing interests: The authors declare no conflict of interest. Funding: Not applicable. Author Contributions: D.S.K. and T.K. conceived of the framework andstructured the whole paper; D.S.K. and T.K. wrote the paper; J.K. and H.Y.K.typed the paper; D.S.K. and T.K.completed the revision of the article. Allauthors have read and agreed to the published version of the manuscript. Acknowledgements: The authors would like to thank Jangjeon Institute forMathematical Science for the support of this research. References 1. Carlitz, L., Degenerate Stirling, Bernoulli and Eulerian numbers, Utilitas Math. (1979), 51–88.2. Bayad, A.; Hamahata, Y., Multiple polylogarithm and multi-poly-Bernoulli polynomials, Funct. Approx. Comment. Math. (2016), part 1, 45–61.3. Hardy, G. H., On the zeroes of certain classes of integral Taylor series. Part II.-On theintegral function formula and other similar functions, Proc. London Math. Soc. (2) (1905), 401–431.4. Hardy, G. H., On the zeroes certain classes of integral Taylor series. Part I.-On theintegral function formula, Proc. London Math. Soc. (2) (1905), 332–339. TAEKYUN KIM , DAE SAN KIM , HAN YOUNG KIM , AND JONGKYUM KWON ∗ 5. Dolgy, D. V.; Kim, D. S.; Kim, T.; Komatsu, T., Barnes’ multiple Bernoulli and poly-Bernoulli mixed-type polynomilas, J. Comput. Anal. Appl. (2015), no. 5, 933-951.6. Kim, D.; Kim T.; J.-J. Seo; Komatsu, T., Barnes’ multiple Frobenius-Euler and poly-Bernoulli mixed- type polynomials, Adv. Difference Equ. (2014), Paper no 92, 16pp.7. Kurt, B. ; Simsek, Y., On the Hermit based Genocchi polynomials, Adv. Stud. Contemp.Math. (Kyungshang) (2013), no. 1, 13–17.8. Kim, D. S.; Kim, T., A note on polyexponential and unipoly functions, Russ. J. Math.Phys. (2019), no. 1, 40-49.9. Kim, D. S.; Kim, T., A note on degenerate poly-Bernoulli numbers and polynomials, Adv. Difference Equ. (2015), 2015:258.10. Kim, M.-S.; Kim, T., An explicit formula on the generalized Bernoulli number with order n , Indian J. Pure Appl. Math. (2000), no. 11, 1455-1461.11. Kim, T.; Kim, D. S., Degenerate polyexponential functions and degenerate Bell polyno-mials, J. Math. Anal. Appl. (2020), no. 2, 124017.12. Kim, T.; Kim, D. S.; Kim, H. Y.; Jang, L.-C., Degenerate poly-Bernoulli numbers andpolynomials, Informatica A note on degenerate Genocchi and poly-Genocchi numbers and polynomials, J. Inequal. Appl. (2020), 2020:110.14. Kim, T.; Kim, D. S.; Kwon, J.; Lee, H., Degenerate polyexponential functions andtype 2 degenerate poly-Bernoulli numbers and polynomials, Adv. Difference Equ. (2020),2020:168.15. Kim, T.; Kim, D. S.; Lee, H.; Kwon, J., Degenerate binomial coefficients and degeneratehypergeometric functions, Adv. Difference Equ. (2020), 2020:115.16. Lewin, L., Polylogarithms and associated functions. With a foreword by A. J. Van derPoorten, North-Holland Publishing Co., New York-Amsterdam, 1981.17. Qi, F.; Kim, D. S.; Kim, T.; Dolgy, D. V., Multiple poly-Bernoulli polynomials of thesecond kind, Adv. Stud. Contemp. Math. (Kyungshang) (2015), 1–7.18. Waldschmidt, M., Multiple polylogarithms: an introduction. Number theory and discretemathematics, (Chandigarh, 2000), 112, Trends Math., Birkh¨auser, Basel, 2002. Department of Mathematics, kwangwoon University, seoul 139-701, Republic ofKorea E-mail address : [email protected] Department of Mathematics, Sogang University, seoul 121-742, Republic of Ko-rea E-mail address : [email protected] Department of Mathematics, kwangwoon University, seoul 139-701, Republic ofKorea E-mail address : [email protected] Department of Mathematics Education and ERI, Gyeongsang National Univer-sity, Jinju 52828, Republic of Korea E-mail address ::